No — but they're related. Icicles hanging off your gutters mean meltwater is reaching the cold eave, which is the same condition that creates ice dams. An ice dam itself is the ridge of ice that forms AT the eave and traps water behind it. You can have icicles without a dam, and a dam without obvious icicles.
Here's the tell: if you see thick ice building up along the eave line — not just icicles dangling from the gutter — you likely have an ice dam, and water is probably backing up behind it. Heavy icicles are still worth paying attention to, because they signal the melt-and-refreeze cycle is running. If you're not sure what you're looking at, send us a photo.